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The creation of information
Assume that the only thing you know about a man with two kids is that at least one of the kids is a daughter. What is the probability that the other kid is a daughter as well? (Boys and girls are assumed to be born equally often.) After the first impulse ("1/2 of course!"), it becomes clear that it is only 1/3. The problem can be mapped to a situation where from the multitude of families with two children, only those with M/M are ruled out, while the equally often cases F/F, F/M and M/F remain, making F/F only one third of all remaining cases. But now, meet Mr. Smith. I don't know much about him (except that he has two children), but when he approached me, he told me: "I am so happy! Victoria just got the scholarship she wanted!" Now what is the probability that Victoria has a sister? Since I only know that Mr. Smith has two children, and one is obviously a girl, I am tempted to map this onto the two-daughter-problem, leading to the answer "1/3". But wait! What if I ask Mr. Smith first, if Victoria is his elder daughter? Assume his answer is yes (and ignore any problems with twins - even then one is typically a few seconds "older" than the other). So now I know that from the cases (F/F, F/M, M/F), M/F also drops out. And now, the probability for F/F just rose to 1/2. Okay, but what if his answer is no? Then Victoria is the younger one, and F/M drops out. Again, the probability rises to 1/2. So I'm going to just ask him: "Well, Mr. Smith, is Victoria your elder daughter? Wait - don't answer, because whatever you may answer, it doesn't matter. The probability just rose from 1/3 to 1/2." Or, even better, I do not even have to ask him, just thinking about the question will shift probabilities to 1/2, which means that the original probability for Victoria having a sister must already have been 1/2. But then the mapping to the two-daughter-problem is obviously false. ---- Suppose you flip a coin that you know nothing about it's fairness twice. Knowing that the first flip is heads tells you no information about the second flip. Knowing that the second flip is tails tells you no information with the first flip. Knowing that at least on of the one of the first two flips is heads tells you no information with the other flip. Knowing that at least on of the one of the first two flips is tails tells you no information with the other flip. Knowing that at least one of the first two flips is heads tell you no information about the first flip. Knowing that at least one of the first two flips is heads tell you no information about the second flip. Knowing that at least one of the first two flips is tails tells you no information about the first flip. Knowing that at least one of the first two flips is tails tells you no information about the second flip. If you Know that at least one of the first two flips is heads and one of the first two flips is tails and the first flip is heads how do you know the second flip is tails?